Method for hypersurface construction in N dimensions

ABSTRACT

A method for determining boundary hypersurfaces from data matrices includes identifying intermediate hypersurfaces, situated between two respective matrix elements, that correspond to at least a portion of at least one boundary hypersurface to be determined. The identified intermediate hypersurfaces are represented by points that are adjacent to the intermediate hypersurfaces. The points that are adjacent to the intermediate hypersurfaces are connected by at least one respective closed curve. Hypersurface components formed by the closed curves are combined to form at least one boundary hypersurface.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a U.S. National Phase application under 35 U.S.C.§371 of International Application No. PCT/EP2012/058873, filed May 14,2012, and claims priority from German Patent Application No. DE 10 2011051 203.9, filed Jun. 20, 2011, and from German Patent Application No.DE 10 2011 050 721.3, filed May 30, 2011. The International Applicationwas published in German on Dec. 6, 2012 as WO 2012/163657 A1 under PCTArticle 21.

FIELD

The invention relates to a method for determining boundary hypersurfacesfrom data matrices. The invention further relates to a device fordetermining boundary hypersurfaces from data matrices.

BACKGROUND

In many technical fields and in research, measurements are obtained as afunction of various parameter values. In this context, individual, aplurality or all of the parameters on which the measurement valuedepends are varied in succession, and the average measurement value isdetermined as a function of the respective set of parameters. So as notto allow the measurement period to increase excessively, someparameters—in particular the parameters which only have a slightinfluence and/or are irrelevant in the respective field ofapplication—are kept at a fixed value and not varied. If for example nparameters are varied, an n-dimensional value matrix is obtained. Inthis context, each determined measurement value is stored in therespective matrix element.

So as to be able to set up analytical and/or numerical models using the(raw) measurement values obtained in this manner or to be able to put inplace further calculations, it is generally necessary initially toconvert these (raw) data into a different format. A conversion which isoften used in this connection involves determining boundary contours (intwo dimensions a boundary line, in three dimensions a boundary surface)which form a boundary between measurement values below a particularvalue and measurement values above this particular measurement value.

A particularly clear application for determining boundaries of thistype, which is commonly made use of in practice, occurs in relation toimage data which are obtained for example by tomography methods(although the type of image generation is irrelevant; for example, imagevalues may be obtained using X-rays, nuclear spin methods or ultrasoundmethods). Tomography methods of this type are used in a wide range oftechnical fields.

Perhaps the best known and most widespread application for this is inthe field of medical technology. In this context, the measurement valuesobtained represent particular material properties (especially tissueproperties), such as the density of the tissue. In this context, themeasurement values are typically obtained in three dimensions. The datacan subsequently be represented for example in the form of greyscalerepresentations. Using greyscale representations of this type, it ispossible for example to distinguish different tissue regions from oneanother (for example organs, bones, cysts, tumours, air-filled cavitiesand the like). However, if the data obtained are in particular toundergo further automated processing and/or be used for purposes otherthan simple observation, it is often necessary to carry out automatedcalculation of the boundary surfaces (in the case of three-dimensionaldata), with as little user intervention as possible.

In the prior art, what is known as the “marching cubes algorithm”,developed in the mid-eighties by W. E. Lorensen and H. E. Cline, is madeuse of for this purpose (see W. E. Lorensen and H. E. Cline, “MarchingCubes; A High Resolution ED Surface Construction Algorithm”, Comput.Graph. Vol. 21 (1987), pages 163-169). This algorithm involvessuccessive migration through the three-dimensional grid. If thealgorithm establishes that for two mutually adjacent 3D grid points therespective measurement value is above the boundary value in one case andbelow the boundary value in one case, the algorithm concludes from thisthat a boundary surface should be arranged here. This determination, asto whether and if applicable where a boundary surface should bearranged, is carried out from a (varying) central grid point in relationto all of the surrounding grid points. With the “collected” knowledge asto in which corner regions or edge regions a boundary surface is to bearranged, the most suitable base boundary surface arrangement (in otherwords the most suitable “template”) is selected from a set of baseboundary surface arrangements (known as “templates”). This check iscarried out for all of the grid points in succession. Subsequently, thebase boundary surface arrangements determined in this manner areinterconnected to give complete boundary surfaces.

Even though the “marching cubes” algorithm often determines helpfulboundary surfaces in practice, there are noticeable and significantproblems with it. A first problem involves selecting the set of“templates”. If this set is selected to be too large, the algorithmgenerally requires too much calculating time. Moreover, template sets ofthis type rapidly become confusing, and templates are therefore easily“forgotten”. A further problem with the “marching cubes” algorithm isthat the surfaces obtained are generally not clear-cut and moreoveroften have holes—that is to say are not completely closed. This can insome cases lead to major problems in the further processing of theboundary surface data. This problem occurs in particular if acomparatively small number of base surface arrangements (templates) areused (for example for reasons of calculating time).

Even though it may still appear possible to implement and use a completeset of base surface arrangements for a program of this type forthree-dimensional data, major problems are encountered as soon asfour-dimensional data are involved (for example 3D data which vary overtime). Accordingly, the “marching cubes” approach cannot de facto begeneralised to general n-dimensional problems where n≧4 (or only withgreat difficulty).

So as to reduce (and ideally to prevent) the above-disclosed problemsinvolving non-closed surfaces, a wide range of proposals for modifyingthe basic “marching cubes” algorithm have already been proposed. Asummary of previous approaches may be found for example in thescientific publication “A Survey of the Marching Cubes Algorithm” by T.S. Newman and H. Yi in Computers & Graphics, Vol. 30 (2006), pages854-879. Improvements to the “marching cubes” algorithm have beendisclosed in the patent literature too, for example in U.S. Pat. No.7,538,764 B2. Even though the improvements disclosed in the prior art tothe original “marching cubes” algorithm provide considerableimprovements, they generally still have drawbacks, in some cases seriousdrawbacks. In particular, in the algorithms known in the art, theproblem of non-closed boundary surfaces is generally still present, inparticular when there are particularly unfavourable measurement valuedistributions.

SUMMARY

In an embodiment, the present invention provides a method fordetermining boundary hypersurfaces from data matrices includingidentifying intermediate hypersurfaces, situated between two respectivematrix elements, that correspond to at least a portion of at least oneboundary hypersurface to be determined. The identified intermediatehypersurfaces are represented by points that are adjacent to theintermediate hypersurfaces. The points that are adjacent to theintermediate hypersurfaces are connected by at least one respectiveclosed curve. Hypersurface components formed by the closed curves arecombined to form at least one boundary hypersurface.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be described in even greater detail belowbased on the exemplary figures. The invention is not limited to theexemplary embodiments. All features described and/or illustrated hereincan be used alone or combined in different combinations in embodimentsof the invention. The features and advantages of various embodiments ofthe present invention will become apparent by reading the followingdetailed description with reference to the attached drawings whichillustrate the following:

FIG. 1 shows an embodiment for the construction of boundary lines in twodimensions;

FIG. 2 shows an embodiment for the construction of a boundary surface inthree dimensions;

FIG. 3 shows examples of different types of initial contact faces fordifferent voxel data value arrangements in three dimensions;

FIG. 4 shows examples of possible ways of processing volume regionswhich contact one another only along a line in three dimensions;

FIG. 5 shows the boundary surfaces illustrated in FIG. 3 in a possibleoptimised form;

FIG. 6 shows the boundary surfaces illustrated in FIG. 5, said boundarysurfaces being reduced to triangles;

FIG. 7 shows different coordinate systems for two dimensions, threedimensions and four dimensions;

FIG. 8 shows the construction of auxiliary points and connecting pointsin accordance with an embodiment of an algorithm for differentdimensions;

FIG. 9 shows the construction of connecting lines between connectingpoints in accordance with a preferred embodiment of an algorithm fordifferent dimensions;

FIG. 10 shows the notation used in a 2*2*2*2 toxel neighbourhood for thedifferent toxels, auxiliary points and connecting points in accordancewith a four-dimensional embodiment;

FIG. 11 is an illustration of the boundary volume of an individual toxelin a 2*2*2*2 toxel neighbourhood in accordance with an embodiment;

FIG. 12 shows the double vectors used for the example shown in FIG. 11in accordance with an embodiment of the algorithm in a four-dimensionalexample environment;

FIG. 13 shows embodiments of generated boundary hypersurfaces in systemshaving different dimensions;

FIG. 14 shows an embodiment of possible ways of handlingfour-dimensional regions of the same type which only contact one anotheralong a surface;

FIG. 15 shows an example of the application of the method tofive-dimensional data; and

FIG. 16 shows a possible method for extracting hypersurfaces in the formof a flow chart.

DETAILED DESCRIPTION

An aspect of the invention provides a method for determining boundaryhypersurfaces from data matrices which is improved over known methods. Afurther object of the invention is to propose a device which is improvedover known devices for determining boundary hypersurfaces from datamatrices.

It is proposed to carry out a method for determining boundaryhypersurfaces from data matrices having the following steps:

-   -   identification of the intermediate hypersurfaces, situated        between two respective matrix elements, that correspond to at        least a portion of at least one boundary hypersurface to be        determined;    -   representation of the intermediate hypersurfaces identified in        this manner by points that are adjacent to intermediate        hypersurfaces;    -   connection of the points that are adjacent to intermediate        hypersurfaces by at least one respective closed curve;    -   combination of the hypersurface components formed by the closed        curves to form at least one boundary hypersurface.

Generally, data matrices are in the form of regular grids, such as inparticular square, cubic etc. grids (using the corresponding, preferablyregular geometric patterns in n dimensions). However, it is alsoperfectly conceivable to use other data structures, such as inparticular irregular grids or grids having different coordinates (forexample spherical coordinates, cylindrical coordinates and the like orthe n-dimensional equivalents thereof). In principle, it does not matterhow many dimensions the data matrix has. It is also possible for thedata matrix to have different dimensions in portions, for example tohave a dimension of n in some sub-regions and a dimension of n+1 or thelike in other sub-regions. Further, the density of individual points inthe data matrix may also vary, in such a way that for example there is ahigher and/or lower density of measurement values in individualsub-regions. The intermediate hypersurfaces may in particular beidentified in such a way that it is checked whether there is a “jump”over the respectively (locally) applicable boundary value between therespectively adjacent matrix elements. For example, there may thus be avalue above the (locally) applicable boundary value in one matrixelement whilst the value in the adjacent matrix element is below the(locally) applicable boundary value. Accordingly, an intermediatehypersurface is to be provided between the two matrix elements. Bycontrast, if there is no “jump” between two adjacent matrix elements, asa matter of basic principle no intermediate hypersurface is arrangedthere. Accordingly, the number of intermediate hypersurfaces per matrixelement may vary. In particular, the number may be between zero and fourin two dimensions, between zero and six in three dimensions, betweenzero and eight in four dimensions, and between zero and ten in fivedimensions (and generally between 0 and 2n intermediate hypersurfaces inn dimensions). In this context, the (geometric) position of theintermediate hypersurface is irrelevant, at least initially. Anarrangement which is particularly advantageous in terms of manageablecalculation is generally provided by an (initially) central arrangementof the intermediate hypersurface between the two respectively adjacentmatrix elements. In mathematical terms, the boundary hypersurface isgenerally a manifold of co-dimension 1. Accordingly, this is generally aline in two dimensions, a surface (for example a square) in threedimensions, a volume (for example a cuboid, cube or the like) in fourdimensions, a tesseract in five dimensions etc. In the following, theintermediate hypersurfaces identified in this manner are represented bypoints that are adjacent to intermediate hypersurfaces. In this context,“adjacent to the intermediate hypersurfaces” may in principle meanbasically any expedient type of arrangement of the relevant points withrespect to the respective intermediate hypersurface. In particular, therespective points may be positioned substantially on (in particularcentrally on) a connecting line between the two adjacent matrix elements(in particular the centres thereof). Some “lateral” variations are ofcourse conceivable, but should preferably not be any greater than wouldcorrespond to the extension of the respective matrix element.Subsequently, the points that are adjacent to intermediate hypersurfacesare connected by at least one closed curve in each case. The type ofline path (initially) basically does not matter. Even though a directconnection may be particularly expedient, it may also be found to beadvantageous if there is at least initially a connection along theboundary edges (in three dimensions; in other dimensions optionally therespectively associated geometric structure). Subsequently, it ispossible (but not compulsory) to carry out further optimisation on therespective connection lines. Subsequently, the hypersurface components(advantageously already optimised at this time) which are formed by theclosed curves are combined to form at least one boundary hypersurface.Depending on the type of data involved, there will be a single (closed)boundary hypersurface or a plurality of (closed) boundary hypersurfaces.If for example there are two separately positioned tumour regions inthree dimensions, it is naturally expedient to calculate two mutuallyindependent boundary hypersurfaces which are closed per se. Moreover,mutually “separate” boundary hypersurfaces may also occur if these arein contact with one another in a lower dimension. Thus, in threedimensions for example it is also possible to refer to mutuallyseparated boundary hypersurfaces even if they touch at a point and/oralong a line. This is also applicable analogously to lower-dimension orhigher-dimension spaces. A major advantage of the proposed method isthat in practice no basic assumptions have to be made. In particular, itis not necessary to provide a set of base hypersurface elementarrangements (known as “templates”). Accordingly, the algorithm isparticularly robust (nearly always resulting exclusively in completelyclosed surfaces which do not have any holes), and is moreoverparticularly simple to generalise to higher dimensions, such as inparticular four dimensions, five dimensions, six dimensions and thelike.

In the method, it is preferred for the points that are adjacent tointermediate hypersurfaces to be, at least in part and/or at least attimes, arranged centred on the intermediate hypersurfaces and/orincluded in the intermediate hypersurfaces in at least one step,preferably in at least a first step. A method of this type is generallyfound to be particularly simple to implement and moreover nearly alwaysto be comparatively fast. Since the points that are adjacent tointermediate hypersurfaces, the curves connecting these points, theresulting hypersurface components and/or the resulting boundaryhypersurfaces (and optionally also other elements) can furthermore beoptimised in a later step, this need not lead to worsening of the endresult. In this context, an arrangement centred on the intermediatehypersurfaces may mean an arrangement along a line connecting thecentres of the corresponding mutually adjacent matrix elements or anarrangement positioned adjacent to this. An arrangement included in theintermediate hypersurfaces may in particular mean an arrangement of therespective point on or adjacent to an intermediate hypersurface (inother words for example a surface in three dimensions and a volume infour dimensions).

In the method, it is further advantageous for at least one closed curveand/or at least one hypersurface component to be optimised, inparticular reduced, particularly preferably locally minimised, at leastat times and/or at least in part. With optimisation of this type, animprovement which is nearly always noticeable in the final boundarysurface(s) can be achieved using comparatively simple and/orcomputationally non-intensive algorithms. The proposed method thusgenerally produces noticeably better results.

A further advantageous embodiment of the proposed method is provided ifat least one point that is adjacent to intermediate hypersurfaces and/orat least parts of at least one curve connecting points that are adjacentto intermediate hypersurfaces and/or at least one hypersurface componentand/or at least one part of at least one boundary hypersurface areplaced and/or displaced as a function of the data values of the adjacentdata matrix elements. By means of an embodiment of this type of themethod, the position of the boundary hypersurface can generally bedetermined even more precisely, by a considerable amount. Purely by wayof example, if for a first measurement value of 51%, a secondmeasurement value of 20% and a boundary value of 50% (resulting in a 50%boundary hypersurface) the final boundary surface is placed preciselymidway between the two mutually adjacent matrix elements, this is nearlyalways less good than if the boundary hypersurface is located close tothe centrepoint of the matrix element having the 51% measurement value.So as to make quantitative displacement possible, basically any knowninterpolation methods may be used (it also being possible to make use offurther data values of matrix elements located nearby), such as linearinterpolation, quadratic interpolation, cubic interpolation, splineinterpolation and the like.

It is further proposed to carry out the method in such a way that atleast one closed curve, preferably a plurality of closed curves, inparticular at least substantially all of the closed curves, aredirectionally orientated and/or bounded by vector-like boundaries atleast at times and/or at least in part. Using directional orientation ofthis type, it is simpler to combine hypersurface components intoboundary hypersurfaces. Moreover, using the directional orientation itis possible to decide on an internal and external region (with respectto the complete boundary hypersurface). In this context, an internalregion may be a region where the data values are above the boundaryvalue, the data values accordingly being below the boundary value in theexternal region (or vice versa). In this context, it is also perfectlypossible to provide directional orientation in a plurality ofdirections. Thus, for example, it is possible for a connecting line inthe form of a pair of mutually antiparallel vectors to be “directionallyorientated”. It is also possible for the directional orientation (inparticular the precise nature and implementation of the directionalorientation) to be dependent on the respective method step and to varyover the course of the method (possible even repeatedly).

A further advantageous development of the method is achieved in that theboundary value which determines the arrangement, at least at timesand/or at least in part, of at least part of at least one boundaryhypersurface is variable at least at times and/or at least in regions,in particular variable by the user, and/or in that the treatment of atleast parts of at least two mutually adjacent boundary hypersurfaces isvariable, in particular variable by the user, at least at times and/orat least in regions. With a development of this type, the proposedmethod can be used particularly universally. The size of a boundaryvalue which is to be selected at least in sub-regions and/or at least inparticular time intervals can thus be adapted flexibly for example by auser of the method. In this case it is for example possible to “prepare”a tumour to be “better for calculation” by way of a correspondinglyselected data value. The variation by the user may either be direct (forexample by inputting a dedicated numerical value) or be transparent tothe user in that he selects for example “bone display”, “tumour display”and the like. The type of treatment of adjacent boundary hypersurfacesmeans in particular whether for example bodies which touch merely at onepoint or on one straight line (and optionally higher-dimensional bodiesdepending on the respective matrix dimension to be dealt with) are to betreated as interconnected bodies or mutually separated bodies.Accordingly, either two mutually separated boundary hypersurfaces or asingle boundary hypersurface in the form of a “thin connecting neck” arethus formed. These numbers of one or two boundary hypersurfaces shouldof course be treated as purely exemplary.

It is further proposed for the method to have at least one method stepin which at least one additional auxiliary structure can be inserted inat least one boundary hypersurface at least at times and/or at least inregions. This may for example take place if particular connection nodeswhich are “physically nonsensical”, “medically nonsensical” or the likeoccur in the “almost finished” boundary hypersurface. In this case,these can subsequently be refined to be physically or medicinally“plausible” by inserting additional auxiliary structures (such asadditional lines and the like). It is also possible for example toachieve a partial and/or optionally (substantially) complete reductionto particular geometrical base structures by inserting auxiliarystructures of this type. For example, it may be found to be expedient tocarry out a (partial and/or substantially complete and/or complete)reduction to triangles and/or tetrahedra (and/or correspondinghigher-dimensional bodies), since structures of this type can forexample be handled particularly well by graphics cards of electroniccomputers, and gains can therefore be made in terms of calculating time.

Moreover, in the method it is particularly expedient for a smoothingstep to be carried out at least at times for at least part of at leastone boundary hypersurface. As a result “physically nonsensical” and“medically nonsensical” (and the like) corners and edges can beprevented and a smoothed boundary hypersurface can be displayed. Theresult of the method can be further improved, in some casesconsiderably.

It is further proposed for the method to be carried out using a datamatrix which is at least three-dimensional, at least four-dimensional,at least five-dimensional, at least six-dimensional, at leastseven-dimensional and/or higher-dimensional at least at times and/or atleast in part. In this context, the aforementioned values may be treatedas discrete individual values, as lower boundary values and/or as upperboundary values. Thus, purely by way of example, a “purelyfour-dimensional” method or a four-to-six-dimensional method should beconsidered to be explicitly disclosed. Moreover, further “discloseddimensions” are 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19 and 20. Inparticular four-dimensional data matrices are extremely expedient formany applications, since they are for example good for modellingthree-dimensional spaces which vary over time. Higher-dimension valuesmay also be found to be valuable, in particular in engineering,metrological and other technical applications as well as in (basic)research work. In this connection, relativistic hydrodynamic simulationcalculations of freeze-out hypersurfaces for multi-particle productionfrom fireballs in heavy ion collisions are one of many examples.Moreover, a further particularly promising field of application, inparticular for four-dimensional methods, is the animated film industry,in which characters which vary over time in three-dimensional cartoonscan be calculated.

In the method, it is particularly expedient for one of the dimensions ofthe data matrix to be time. In this case, an object which varies overtime can be analysed in depth particularly well, and moreover, it ispossible to calculate intermediate steps (in particular by interpolationmethods) and predictions (in particular by extrapolation methods) in aparticularly simple manner.

In the method, it is particularly advantageous for the data of the datamatrix to be measurement values, in particular image values and/orphysical measurement values, particularly preferably medical measurementvalues. In this case, the proposed method may be found to be of aparticularly high relevance in practice.

A device, in particular a computer device, is proposed, which isconfigured and set up in such a way that it carries out a method of thetype disclosed above, at least at times and/or at least in part. Inparticular, the device may be used for determining boundaryhypersurfaces from data matrices. A device of this type may have theadvantages and properties disclosed above at least in an analogousmanner. Moreover, the device may be developed within the meaning of thepresent description (at least in an analogous manner).

To explain the proposed algorithm, the simplest case will initially bedescribed, namely applying the method to measurement values which are inthe form of a two-dimensional matrix. For example, this may be atwo-dimensional computer tomography section through a tissue or throughthe body of a patient.

In this context, the algorithm is described in greater detail referringto the various pictures of FIG. 1. So as further to limit thedescription to the essential, only two different data values are used,namely “dark” and “light”. An example of a data set of this type isshown in FIG. 1a . The 2D grid 2 of the image 1 can be seen, in whichthe different image points 3, 4, specifically the light image points 3and the dark image points 4, are shown.

Initially, in a first method step, all “transition pairs”—at which thereis therefore a contrast transition (transition from a light image point3 to a dark image point 4 or vice versa) on the connecting line (notshown) of the two adjacent image points 3, 4—on the entire 2D grid 2 aredetermined. At contrast points of this type, an intermediatehypersurface 5 is accordingly provided in each case. This is shown inFIG. 1c . In the example presently shown, the intermediate hypersurfaces5 are positioned precisely on the grid lines of the 2D grid 2. Apositioning of this type is advantageous, but is not absolutelycompulsory.

Algorithmically, all “transition pairs” can be detected for example by“running through” all of the image points 3, 4 “in succession”. For thispurpose, the image points 3, 4 may for example be “actuated” insuccession in rows, while checking for the presence of “transitionpairs” for each image point 3, 4 in a corresponding 2*2 image pointneighbourhood. Starting from a particular image point 3, 4, a 2*2 imagepoint neighbourhood is defined for example by the “starting imagepoint”, the right-hand neighbour thereof, the upper neighbour thereofand the upper-right neighbour thereof. Since conventionally only “directtransition pairs” (and not “diagonal transition pairs”) are used for thecontinuing method sequence, it may also be sufficient, for eachindividual image point 3, 4, merely to check for the presence of acontrast transition from the respective right-hand and upper image pointneighbours 3, 4. By way of the proposed mode of operation, the number ofimage point combinations to be checked can be greatly reduced, and thiscan help to save calculating time.

Subsequently, in a second method step, for the individual intermediatehypersurfaces 5 the corresponding auxiliary points 6 are determined (seeFIGS. 1c and 2e ). In the presently shown embodiment, the auxiliarypoints 6 are in each case arranged in the centre of the grid edges ofthe 2D grid 2 or in the centre of the respective intermediatehypersurfaces 5. This positioning corresponds to an arrangement in thecentre of the connecting line between every two matrix element centresof respectively adjacent matrix elements (in other words image points 3,4, auxiliary points 6 only occurring between matrix elements having acontrast transition). Subsequently, the auxiliary points 6 determined inthis manner are each connected to form closed auxiliary curves 14. Theauxiliary curves 14 are passed through the connecting points 29 at whichthe individual intermediate hypersurfaces 5 touch one another. In thecase shown, the closed auxiliary curves 14 correspond to a sequence ofthe intermediate hypersurfaces 5. As will be discussed again in greaterdetail below, in the example shown in FIG. 1c , depending on theselected mode, a total of three closed auxiliary curves 14 (in the“separation mode” FIG. 1e ) or merely two closed auxiliary curves 14 (inthe “connection mode”; FIG. 1f ) are obtained.

As can further be seen in FIG. 1c , the closed auxiliary curves 14 areeach provided with a directional orientation. From the orientation ofthe auxiliary curves 14, it can thus be determined whether the imagepoints 3, 4 positioned inside the auxiliary curve 14 are dark imagepoints 4 or light image points 3. In the presently shown embodiment, thedirection convention is selected in such a way (see also FIG. 1b ) thata closed auxiliary curve 14 extending anticlockwise includes dark imagepoints 4 in the interior thereof (light image points 3 accordingly beingpositioned outside the auxiliary curve 14 extending anticlockwise). Bycontrast, if the closed auxiliary curve 14 is directed clockwise, thismeans that light image points 3 are arranged in the interior of theclosed auxiliary curve 14 extending clockwise (and accordingly there aredark image points 4 outside the closed auxiliary curve 14 extendingclockwise). It is noted that the direction convention is selectedarbitrarily and can perfectly well be selected differently.

Further, it is also perfectly possible for further auxiliary curves 14to be included inside an auxiliary curve 14. Accordingly, it is possiblefor dark image points 4 to be positioned inside a closed auxiliary curve14 extending anticlockwise, in the “proximate region” to the auxiliarycurve 14, whilst there are light image points 3 again in a further“inner” auxiliary curve 14, positioned inside the “outer” closedauxiliary curve 14 and having a direction extending clockwise. These“substructures” may also be nested repeatedly. A simple example of“substructures” of this type is shown in FIG. 1 (in particular FIG. 1c).

The auxiliary curves 14 are subsequently optimised in that the auxiliarypoints 6 (optionally after positional correction) are directlyinterconnected, in other words the connecting points 29 are omitted.This leads to the length of the auxiliary curves 14 being minimised(local minimum in each case), ultimately resulting in the closed curves7, 8 shown in FIG. 1e . The closed curves 7 are directed anticlockwise,and thus contain dark image points 4 in the interior thereof.Accordingly, the closed curves 8 are directed clockwise, and thuscontain light image points 3 in the interior thereof.

A further special case occurs if two sets of image points (two sets eachconsisting of dark image points 4 in the embodiment shown in FIG. 1c )touch at merely a single contact point 9. In this case, it may beexpedient for the respective sets of image points 4 to be treated asseparate sets (ultimately resulting in the total of three closed curves7, 8 shown in FIG. 1e ). However, it may also likewise be expedient totreat the sets of image points which touch at a single contact point 9to belong together, ultimately resulting in the situation shown FIG. 1f, in which merely two closed curves 7, 8 are produced by the algorithm.It is advantageous for the decision as to whether sets of image pointsof this type should be treated as cohesive or as separate from oneanother to be left to the user, as explained above. Said user may forexample define what actions should be taken in a case of this typebefore the closed curves 7, 8 are constructed (for example by selectinga corresponding button on a computer screen).

Contact points 9 of this type are identified and handled using the logicshown in greater detail in FIG. 1d . For this purpose, the initialauxiliary curves (corresponding to the interconnected intermediatehypersurfaces 5) are preferably already provided with a directionalorientation (same convention as the directional orientation of theclosed curves 7, 8; see FIGS. 1b, 1c, 1e, 1f ). Following the sequenceof the auxiliary curves 14 or the directed intermediate hypersurfaces 5,an auxiliary curve 14 or an intermediate hypersurface 5 having theopposite direction is encountered at a contact point 9 (see FIG. 1d ).If the setting is such that sets of points which touch at a singlecontact point 9 are to be combined (FIG. 1f ), there is a deflection “tothe right” in quadrant IV. By contrast, if the sets of points are to beseparated from one another, there is a deflection “to the left” inquadrant I. The same applies when coming from the direction of quadrantII/III.

Referring to FIGS. 2 to 6, the application of the proposed method in thecase of three-dimensional data will be explained.

For simplicity, FIG. 2 shows a single 3D volume element 10 (known as avoxel), embedded in a 3D grid 12 which is merely sketched in. The voxel10, which is cubic in this case, is a dark voxel 10 which is enclosed byexclusively light voxels 11. The dark voxel 10 thus contacts an adjacent(light) voxel 11 at each of a total of six faces 13. The initialrelationships are shown FIG. 2a . Since this is a single dark voxel 10in the middle of otherwise exclusively light voxels 11, it has to becompletely enclosed by a boundary hypersurface 17 after the algorithm isrun (cf. FIG. 2f ).

Since there is a contrast (between dark and light) between the darkvoxel 10 shown and each of the light voxels 11 surrounding it, anintermediate hypersurface 13 is to be arranged between the dark voxel 10and each of the six surrounding light voxels 11. In this case, saidhypersurface is identical to the faces 13 of the cubic voxel 10 or thefaces 13 of the 3D grid 12.

In this context—similarly to what was explained previously in twodimensions—the contrast transitions can be identified by passing throughall of the voxels 10 of the data matrix in succession, the presence ofcontrast transitions being searched for in a 2*2*2 neighbourhood in eachcase.

Subsequently, in a further step, an auxiliary point 6, centrallyarranged in each case in the embodiment shown, is assigned to each ofthe faces 13. These auxiliary points 6 are in turn interconnected by aplurality of auxiliary curves 14, which are each closed per se. Of theseauxiliary curves 14 which are closed per se (each “assigned” to thecubic voxel 10), and of which there are a total of eight in theembodiment shown, only one auxiliary curve 14 is shown in FIG. 2c forreasons of clarity (the other auxiliary curves 14 “framing” theremaining corners of the voxel 10). As can be seen from FIG. 2c , theauxiliary curve 14 is selected in a directionally orientated manner,applying the direction convention used previously in two dimensions. Thedirection indicates that the dark voxels 10 are located inside and thelight voxels 11 are located outside. The closed auxiliary curves 14 canbe generated with particularly simple programming in that the auxiliarypoints 6 are in each case interconnected by a connecting line 15 (seeFIG. 2b , where for reasons of clarity only a single connecting line 15is provided in greater detail), and this connecting line 15 isrepresented in the form of mutually antiparallel vectors (“doublevectors”) (the other connecting lines 15 extend through the remainingside edges of the voxel 10). An auxiliary curve 14 which is closed perse is then generated by placing direction vectors having the samedirectionality in sequence.

In a further step, the auxiliary curves 14 are subsequently simplifiedin that the intermediate points on the side edges of the voxel 10 areremoved from the curve path. This results in the triangle 16 visible inFIG. 2d . The triangle 16 produces a local minimum surface, in otherwords is locally minimised in terms of the surface area and perimeterthereof. Since not only the regions of the voxel 10 shown at the frontright and top in FIG. 2 are processed by the algorithm, seven furthertriangles are generated aside from the triangle 16 shown in bold in FIG.2d , and together form an octahedron 17 (or form an octahedral surfacewhich encloses an octahedron 17). The octahedron 17 is shown in the formof a gridline model in FIG. 2e . In FIG. 2f , in addition to the simplegrid model, the octahedron 17 is further shown together with thedirectional orientation shown in FIG. 2c of the auxiliary curves 14(which is also maintained for the triangles 16). On the basis of theperipheral direction of the individual triangles 16, normal vectors 18,in other words directionally orientated surface perpendiculars (surfacenormals), can be defined for each triangle 16. These result in the“inside” and “outside” of the boundary hypersurface, thus in this caseof the octahedron 17.

In general, of course, more complex data arrangements than thearrangement shown in FIG. 2 need to be processed by the proposedalgorithm. In FIG. 3, purely by way of example, the intermediate resultscorresponding to FIG. 2c are shown for some arrangements of dark voxels10 and light voxels 11 (merely the boundary faces at which there is acontrast transition being shown in FIG. 3). In this context, for some ofthe topologies shown this results in configurations similar to thesingle contact point 9 in FIG. 1c . This is the case in FIGS. 3e and 3g. Since this is now a problem one dimension higher, it is expedient notto provide the possibility of (optionally) connecting or separatingmutually contacting sets of light voxels 11 and dark voxels 10 whenthese sets merely contact one another at a single point, but only whentwo sets of the same type contact one another along a line. This is thecase in FIG. 3e , where at the front left there is a column of twooverlapping dark voxels 10 and at the rear top right there is a singledark voxel 10. The “dark column” and the single dark voxel 10 contactone anther along a contact line 19, and this is shown by way of acircular line at the midpoint of the contact line 19 (the circular lineis supposed to symbolise a possible alternative processing in a“separation mode” or in a “connection mode”). A situation of this typeoccurs in FIG. 3g , where two dark voxels 10 are arranged along a“diagonal” in such a way that in this case two a contact line 19 isgenerated. Purely for completeness, it is noted that FIG. 3 merely showsindividual exemplary arrangements of dark 10 and light voxels 11.

FIG. 4 explains in greater detail how the algorithm generates theresulting boundary hypersurfaces when there is a contact line 19. Inaccordance with the methods proposed herein, the two sets of dark voxels10 (top left and bottom right in FIG. 4a ) are separated or connected(for example depending on a user specification) during the reductionstep when simplifying the auxiliary curves 14 (see transition from FIG.2c to FIG. 2d ). The “logic” used by the algorithm is shown inparticular in FIG. 4b . FIG. 4b shows the handling of the differentdirection vectors (four drawings in total) in accordance with thesituation shown in FIG. 4a . Depending on whether the “connection mode”or the “separation mode” is selected, either the elbowed white arrow(corresponding to the “separation mode”) or the elbowed black arrow(corresponding to the “connection mode”) is followed. Accordingly, inthe “separation mode” this results in the boundary lines 20 shown inFIG. 4c and in the two mutually separated octahedra 17 (or octahedralsurfaces FIG. 4e ) or the boundary lines 20 shown in FIG. 4d and thusultimately in the overall volume 21 shown in FIG. 4f (or the surface ofthe body shown in FIG. 4f ).

Finally, FIG. 5 shows the topologies shown in FIG. 3 in a form afterfurther processing, specifically after the auxiliary lines 14 have beensimplified into surface components 22. This corresponds to thetransition from FIG. 2c to FIG. 2d . In this context, the “connectionmode” was used in the case of the topologies illustrated in FIGS. 3e and3 g.

A further processing option for the resulting hypersurface components22—and thus ultimately for the overall hypersurface—involves reducingall of the faces to triangles by introducing additional auxiliary lines23 (see FIG. 6). On the one hand, triangles of this type can generate asmoother and more homogeneous overall surface. A further advantage is inparticular the option of further processing or visualisation of the dataon an electronic computer, since modern graphics cards can be addressedin a particularly simple manner using a number of individual triangles,the graphics card itself calculating the spatial positioning and theresulting light, colour and other spatial effects. The central processor(CPU) can thus be unburdened considerably and/or the results can bevisualised much more rapidly. In between, it may be found to beexpedient for parts of the calculations required for the method to beshifted from the CPU (central processor) to the GPU (graphics processingunit).

FIGS. 7 to 9 show the transition of the algorithm from two-dimensionalthrough three-dimensional to four-dimensional data.

In FIG. 7, FIG. 7a shows a single dark image point 4 in a 2D grid 2. InFIG. 7b , a single dark voxel 10 can be seen in a 3D grid 12. Finally,FIG. 7c shows the analogous four-dimensional relationships, in otherwords a dark tesseract 24 (a “four-dimensional cube”) in afour-dimensional grid 25. The axes of the four-dimensional grid 25 arelabelled X, Y and Z for the spatial axes and t for the fourth dimension(for example time). The type of display of the tesseract 24 in FIG. 7ccan be thought of as a voxel which varies over time. The representationof the tesseract in FIG. 7c is supposed to symbolise this variation of avoxel over time. As is known from mathematics, a two-dimensional object(in FIG. 7a a square) is delimited by one-dimensional objects, in thiscase by four lines. By contrast, a volume element (cubic voxel 10 inFIG. 7b ) is delimited by surface elements, in this case by foursquares. The four-dimensional tesseract in FIG. 7c is accordinglydelimited by three-dimensional bodies, in this case by eight cubes.

FIG. 8 shows a preferred construction of intermediate hypersurfaces 5,13 and auxiliary points 6 in two dimensions (FIG. 7a ), three dimensions(FIG. 7b ) and four dimensions (FIG. 8c ). For reasons of clarity, therespective construction of the intermediate hypersurfaces 5, 13 and ofthe auxiliary points 6 is only shown in one position in each case.

In FIG. 8a , a contrast transition 26 represented by an arrow (arrowdirection from “dark” to “light”) is shown in a two-dimensional grid. Inthe case shown, the left image point is a dark image point 4, whilst theright image point is a light image point 3. The direction of thecontrast transition arrow 26 goes from “dark” to “light”. As explainedpreviously in connection with FIG. 2, as a result of the contrasttransition 26 an intermediate hypersurface 5 (in the present 2D case aline) is provided. The auxiliary point 6 is arranged in the centre ofthe 2D intermediate hypersurface 5.

The relationships in three dimensions are shown in FIG. 8b . Thecontrast transition 26 from a dark voxel (left of the image) to a lightvoxel 11 (right of the image) is likewise represented by a contrasttransition arrow 26. An intermediate hypersurface 13 (in threedimensions a surface 13, in this case in the form of a square) islikewise arranged between the two different voxels 10, 11. A (centrallyarranged) auxiliary point 6 is arranged in the centre of theintermediate hypersurface 13.

In a corresponding manner, in four dimensions (see FIG. 8c ) aninterposed intermediate hypersurface 28 is also provided between twodifferent tesseracts 24, 27 (dark tesseract 24 and light tesseract 27)and in this case is in the form of an intermediate cube 28. Analogouslyto in two dimensions and in three dimensions, in four dimensions too anauxiliary point 6 is provided in the centre of the intermediate cube 28.

Subsequently, in FIG. 9, the construction of the auxiliary curves 5, 14in different dimensions is explained. As is already known from thepreviously explained two-dimensional and three-dimensional cases, theauxiliary points 6 are used as starting points for the construction ofthe hypersurfaces (optionally in different method stages) which are tobe generated. In two dimensions (FIG. 9a ), the connecting points 29 areshown at which two intermediate elements 5 or (diagonally) adjacent gridelements touch one another in each case. In this context, the connectingvectors between two auxiliary points 6 to be interconnected are passedthrough the connecting points 29. In a further step, the connectingpoints 29 are removed (during an optimisation step), resulting in adirect connection between two adjacent connecting points 6 and thusoptimising the resulting overall hypersurface.

FIGS. 9b and 9c show the relationships for three dimensions. In thiscase too, the auxiliary points 6 to be interconnected are interconnectedvia connecting points 29 using connecting lines 15 in the form of doublevectors (see in particular FIG. 9). After the double vectors have beeninterconnected in each case to form closed auxiliary curves 14 (cf. FIG.2c ), “raw hypersurfaces” are present, which represent manifolds,already different, of inclusive curves, but expediently are furtheroptimised.

The relationships in four dimensions are explained in greater detail inFIG. 9d to f . FIG. 9d shows an intermediate cube 28. In this case too,connecting lines 25 in the form of double vectors (see FIGS. 9e and 9f )are guided to the auxiliary points 6 of the adjacent intermediate cubes28, starting from the auxiliary point 6 located in the centre, via theconnecting points 29, of which there are six in four dimensions. FIG. 9eis an exploded drawing of the intermediate cube 28 shown cohesively inFIG. 9d , it being possible to see the total of eight intermediate cubeeighths 48. In FIG. 9f , the direction of the contrast transition 26 isreversed with respect to FIG. 9e (the contrast transition arrow 26points in the opposite direction). Therefore, in FIGS. 9e and 9f , adistinction is made between, one the one hand, the case marked “plus”(FIG. 9e ), in which the contrast transition extends in the positive X,Y, Z or t direction, and, on the other hand, the case marked “minus”(FIG. 9f ), in which the contrast transition 26 extends in the negativeX, Y, Z or t direction. In four dimensions, there is a larger number ofconnecting lines 15.

The following discussion relates to an individual 2*2*2*2 toxelneighbourhood (“4D neighbourhood cell”) within the four-dimensionalgrid, said neighbourhood likewise being represented by a tesseract inthis case. In this context, a “toxel” is a four-dimensional hyper-volumeelement, thus corresponding to a voxel (in three dimensions) or a pixel(in two dimensions). Within a 4D neighbourhood cell, up to sixteentesseracts may be either dark 24 or light 27. Thus, within a 4Dneighbourhood cell, each of these sixteen tesseracts is only representedby one sixteenth of the total hypervolume of the 4D neighbourhood cell.Within a 4D neighbourhood cell, an individual tesseract 24, 27 thus hasonly four closest neighbours, with which it shares one eighth of anintermediate cube 28 (in other words one intermediate cube eighth 48) asa contact volume. In total, this results in 192 different paths on whichthe auxiliary points 6 are to be interconnected via intermediateconnecting points 29.

The possible triplets of different paths are listed in Table I. In thiscontext, the notation explained in FIG. 10 is used. FIG. 10a initiallyshows an individual 4D neighbourhood cell (a four-dimensional gridelement). A 4D neighbourhood cell of this type is obtained by proceedingfrom a particular toxel (“starting toxel”) in one direction in each caseout of the total of four spatial dimensions (but not in the respectivecounter direction). The sixteen individual toxels of this 4Dneighbourhood cell are labelled with the numbers 0-15. In this context,the numbers from 0 to 7 inclusive denote the “past”, whilst the numbersfrom 8 to 15 inclusive describe the “future”. FIG. 10b shows, numberedin order, the auxiliary points 6 of the volumes (intermediate cubes 28)delimiting the individual toxels. These correspond to the columnlabelled “Center ID” in Table I. In turn, FIG. 10c shows the connectingpoints 29 for mutually adjacent intermediate cubes 28 along with therespective numbering thereof. The numbers used in the paths in Table Iand the significance thereof may be derived in particular from FIGS. 10band 10c . In FIG. 10d , by way of illustration, an example comprisingfour contrast transition vectors 26 is shown. In this context, by way ofexample, a toxel labelled with number “0”, having a positive (“plus”)orientation in the X, Y, Z and t directions, is arbitrarily singled out.In this context, the four contrast transitions 26 (in the X, Y, Z and tdirections) are superposed using the auxiliary points 6, which are in arelationship with one another and are arranged in the centre of theconnecting cube 28.

FIG. 11 shows the four boundary volumes 31, 32, 33, 34 (in other words,in the embodiment shown in this case, the eighths 48 of an intermediatecube 28) for an individual toxel labelled 7 (see FIG. 10a ). FIG. 11ainitially shows all four eighths 48 of an intermediate cube 28(specifically the intermediate eighths 31, 32, 33, 34) within a 4Dneighbourhood cell. FIGS. 11 b, c, d and e show the individual eighths48 of the intermediate cube 28 again individually in greater detail.FIG. 11b shows volume element number 11+ (denoted by reference numeral31 in FIG. 11) in the positive X direction. FIG. 11c shows volumeelement number 9− (denoted by reference numeral 32 in FIG. 11) in thenegative Y direction. FIG. 11d further shows volume element number 6−(denoted by reference numeral 33 in FIG. 11) in the negative Zdirection, and finally FIG. 11e shows volume element number 18+ (denotedby reference numeral 34 in FIG. 11) in the positive t direction.

Finally, FIG. 12 continues FIG. 11, which was described above. FIG. 12aillustrates the four initial cyclic vector paths. These extend along thepoints 37→11→36→6→34→9→37, along the points 36→11→49→18→44→6→36, alongthe points 47→18→49→11→37→9→47, and along the points 34→6→44→18→47→9→34.In this context, it should be noted that all of the points having anumber ≧32 are connecting points 29 which are deleted in an optimisationstep. After this optimisation step, this ultimately results in theoptimised cyclic vector path 11→6→9→11 shown in FIG. 12b (along with thetetrahedron embedded in the four-dimensional tesseract) for the firstvector path, the optimised cyclic vector path 11→18→6→11 shown in FIG.12c (including the resulting tetrahedron) for the second vector path,the optimised cyclic vector path 18→11→9→18 shown in FIG. 12d (includingthe optimised tetrahedron) for the third vector path, and the optimisedcyclic vector path 6→8→9→6 shown in FIG. 12e (including the optimisedtetrahedron) for the fourth vector path. FIG. 12f shows the optimisedtetrahedron again in a drawing without any vector paths. For reasons ofclarity, the edges have not been drawn in the form of antiparallelvectors (although these are still present as before).

Finally, this results in the boundary hypersurface shown in FIG. 13c ,in the form of sixteen tetrahedra, since overall the proposed algorithmprocesses all possible positionings of an individual dark tesseract 24which may occur within a 4D neighbourhood cell. In this context, thetetrahedra are vectorially directed, although this is not shown ingreater detail for reasons of clarity. For comparison, FIG. 13 alsoshows the two-dimensional case (FIG. 13a ) and the three-dimensionalcase (FIG. 13b ).

In four dimensions too, the problem previously encountered in two andthree dimensions still occurs as to how to proceed when dark regions 24or light regions 27 do not contact one another in all n−1 dimensions (inother words only have an (n−2)-dimensional hypersurface and/or an evenlower-dimension hypersurface in common; see FIG. 1c or FIG. 4). In fourdimensions, in this context it is additionally possible to distinguishthe cases where two adjacent toxel elements touch one another within avolume (in other words have a surface in common), touch one anotheralong a surface (in other words have a line in common) or touch oneanother along a line (in other words have a point in common).Analogously to the three dimensional case, it is inexpedient to have twocorresponding regions (in other words two light regions or two darkregions) defined as interconnected if they touch one another along asurface (line in common) or merely along a line (point in common) infour dimensions.

FIG. 14 shows the situation within a 4D neighbourhood cell where twotoxels (in this case tesseracts) “of the same type” have a contactsurface 36 in common. The contact surface 36 can be seen in FIG. 14a inthe contact cube 35. FIG. 14b shows the initial connecting lines 15which extend between the individual auxiliary points 6 via theconnecting points 6. As disclosed previously and shown in FIG. 14b ,these connecting lines are positioned in the manner of antiparalleldouble vectors. Depending on whether a “connection mode” or a“separation mode” is selected, the corresponding vector is suitably“elbowed” in the region of the contact surface 36 and continued. This isshown in FIG. 14c . If the “connection mode” is selected, the resultingauxiliary curve 14 follows the path illustrated in the form of a blackarrow. By contrast, if the “separation mode” is active, the auxiliarycurve 14 follows the path illustrated by way of a white double-headedarrow.

FIG. 14d shows the result when the “separation mode” is used, whilstFIG. 14e shows the result when the “connection mode” is used.

For completeness, it is noted that when the presently proposed preferredembodiment of an algorithm is used in the case of two toxels “of thesame type”, which merely have one connecting line or merely a singlepoint in common, the “separation mode” is automatically used. Of course,a different mode of operation is also possible.

Of course, it is also possible to reduce the boundary volumes totetrahedra in four dimensions—analogously to in three dimensions, wherethe boundary surface is reduced to triangles. This may also havenumerical advantages and/or speed advantages, in particular if graphicscards having their own coprocessors are used.

Finally, FIG. 15 also shows in brief the application of the proposedalgorithm for a five-dimensional space. The embodiment shown in FIG. 15ainvolves is a five-dimensional hypercube, known as a penteract 37. As isknown from mathematics, a penteract 37 is delimited by ten tesseracts38. In this context, FIG. 15b illustrates the contact tesseract 38 whichforms the connection between two penteracts 37. In this context, thepenteract 37 illustrated on the left of the drawing is a “dark”penteract, whilst the penteract 37 shown on the right is a “light”penteract. There is thus a contrast, represented by the contrasttransition arrow 26, between the two penteracts 37 of FIG. 15.Analogously to in lower dimensions (the above description havingdescribed in detail the two-dimensional, three-dimensional andfour-dimensional cases), in this case too the connecting hyperelements(in this case a tesseract 38) positioned along the contrast transition26 are initially reduced to the centrally positioned auxiliary point 6thereof. Starting from the centrally positioned auxiliary point 6,connecting lines in the form of antiparallel vectors to the respectivelyadjacent auxiliary points 6 are drawn via the respective connectingpoints 29.

Finally, FIG. 16 further shows a hypersurface extraction method 39 in aschematic flow chart. Initially, the data which are in matrix form (thematrices being of a corresponding dimension such as a dimension of four)are read 40 into the corresponding device. The device may for example bea conventional commercial, program-controlled computer. Preferably, thecomputer comprises a plurality of processors, since the proposed methodcan be run in parallel, in such a way that a significant time gain canbe made when calculating the hypersurfaces.

Subsequently, the method starts by identifying 41 (“identifyinginterposed intermediate hypersurfaces”) preliminary hypersurfaceregions, between which there is a contrast transition, betweenrespectively adjacent matrix elements. In this context, whether there isa sufficient contrast transition can be defined using correspondingparameters. In particular, a boundary value can be defined in such a waythat there is a sufficient contrast transition when the data in onematrix element are above the boundary value and those in the adjacentmatrix element are below the boundary value.

On the basis of the preliminary hypersurface regions which are thusdetermined, the auxiliary points 6 respectively positioned in the centreof the hypersurface regions are calculated 42. Subsequently, in afurther method step 43 the positioning of the connecting points 29 iscalculated. Using the knowledge of the auxiliary points 6 and theconnecting points 29, the connecting lines 15 are subsequently “laidout” in the form of antiparallel vectors and assembled to formhypersurface components (method step 44). Subsequently, in a furthermethod step 45, the hypersurface components determined in this way areinitially optimised, and in a subsequent further method step 46, theoptimised hypersurface components are combined to form an overallboundary hypersurface (or optionally a plurality thereof). The dataobtained in this manner are stored in an output step, after which thehypersurface extraction method 39 ends.

While the invention has been illustrated and described in detail in thedrawings and foregoing description, such illustration and descriptionare to be considered illustrative or exemplary and not restrictive. Itwill be understood that changes and modifications may be made by thoseof ordinary skill within the scope of the following claims. Inparticular, the present invention covers further embodiments with anycombination of features from different embodiments described above andbelow.

The terms used in the claims should be construed to have the broadestreasonable interpretation consistent with the foregoing description. Forexample, the use of the article “a” or “the” in introducing an elementshould not be interpreted as being exclusive of a plurality of elements.Likewise, the recitation of “or” should be interpreted as beinginclusive, such that the recitation of “A or B” is not exclusive of “Aand B,” unless it is clear from the context or the foregoing descriptionthat only one of A and B is intended. Further, the recitation of “atleast one of A, B and C” should be interpreted as one or more of a groupof elements consisting of A, B and C, and should not be interpreted asrequiring at least one of each of the listed elements A, B and C,regardless of whether A, B and C are related as categories or otherwise.Moreover, the recitation of “A, B and/or C” or “at least one of A, B orC” should be interpreted as including any singular entity from thelisted elements, e.g., A, any subset from the listed elements, e.g., Aand B, or the entire list of elements A, B and C.

LIST OF REFERENCE NUMERALS

1 image

2 two-dimensional grid

3 light image point

4 dark image point

5 intermediate hypersurface

6 auxiliary point

7 closed curve (anticlockwise)

8 closed curve (clockwise)

9 contact point

10 dark voxel

11 light voxel

12 three-dimensional grid

13 surface

14 auxiliary curve

15 connecting line

16 triangle

17 octahedron

18 normal vector

19 contact line

20 boundary line

21 overall volume

22 hypersurface component

23 support line

24 dark tesseract

25 four-dimensional grid

26 contrast transition

27 light tesseract

28 intermediate cube

29 connecting point

30 toxel number 7

31 volume element number 11+

32 volume element number 9−

33 volume element number 6−

34 volume element number 18+

35 contact volume

36 contact surface

37 penteract

38 contact tesseract

39 hypersurface extraction method

40 read in data

41 identifying hypersurface regions

42 calculating auxiliary points

43 calculating connecting points

44 laying out auxiliary lines and calculating hypersurface components

45 optimising hypersurface components

46 assembling hypersurface components

47 storing data

48 intermediate cube eighth

What is claimed is:
 1. A method for producing, by a computer devicehaving one or more processors, at least one boundary hypersurface from adata matrix, the method comprising: receiving the data matrix;identifying contrast transitions situated between respective pairs ofmatrix elements; placing, by the one or more processors, at eachidentified contrast transition, an intermediate hypersurface;representing, by the one or more processors, each intermediatehypersurface with a structure including an auxiliary point and one ormore connecting points, wherein the auxiliary point is positioned on aline that connects the corresponding two respective matrix elementsbetween which a contrast transition is situated, and wherein the one ormore connecting points are each positioned on a boundary with anotherintermediate hypersurface; generating, by the one or more processors,one or more vector paths that connect, in a defined sequence to formdirectionally oriented closed curves that form hypersurface components,auxiliary points of intermediate hypersurfaces by extending throughconnecting points between adjacent auxiliary points; combining, by theone or more processors, hypersurface components formed by the one ormore directionally oriented closed curves to form at least one boundaryhypersurface; wherein the data matrix has a dimension n and thehypersurface has a dimension n−1; wherein, if the connecting pointscomprise one or more contact points, a contact point being a connectingpoint having neighboring matrix elements internal to the hypersurfacewhich contact one another in at most n−2 dimensions, the one or moreprocessors generate the vector paths by selecting one of a connectionmode or a separation mode, such that in the connection mode, the one ormore processors generate vector paths to include the contact points inthe region internal to the hypersurface, and in the separation mode, theone or more processors generate the vector paths to exclude the contactpoints from the region internal to the hypersurface; and storing datarepresentative of the at least one boundary hypersurface.
 2. The methodaccording to claim 1, wherein the each auxiliary point of the structurerepresenting a respective intermediate hypersurface is located at thecenter of the respective intermediate hypersurface.
 3. The methodaccording to claim 1, wherein at least one closed curve and/or at leastone hypersurface component is optimized, at least at times and/or atleast in part.
 4. The method according to claim 1, wherein at least oneauxiliary point and/or at least parts of at least one of thedirectionally oriented closed curves and/or at least one hypersurfacecomponent and/or at least one part of at least one boundary hypersurfaceare placed and/or displaced as a function of the data values of adjacentdata matrix elements.
 5. The method according to claim 1, wherein theidentifying contrast transitions situated between respective pairs ofmatrix elements comprises using a boundary value.
 6. The methodaccording to claim 5, wherein the boundary value is variable by a user.7. The method according to claim 1, further comprising inserting atleast one additional auxiliary structure in at least one boundaryhypersurface at least at times and/or at least in regions.
 8. The methodaccording to claim 1, further comprising carrying out a smoothing stepat least at times for at least part of at least one boundaryhypersurface.
 9. The method according to claim 7, further comprisingcarrying out a smoothing step at least at times for at least part of atleast one boundary hypersurface.
 10. The method according to claim 1,wherein the data matrix has a number of dimensions between three andtwenty.
 11. The method according to claim 1, wherein one of thedimensions of the data matrix is time (t).
 12. The method according toclaim 10, wherein one of the dimensions of the data matrix is time (t).13. The method according to claim 1, wherein data included in the datamatrix includes measurement values.
 14. The method according to claim 3,wherein the at least one closed curve and/or at least one hypersurfacecomponent is reduced.
 15. The method according to claim 3, wherein theat least one closed curve and/or at least one hypersurface componentlocally minimized.
 16. The method according to claim 1, furthercomprising providing a grid structure having lines that define edges ofinterior regions, wherein each interior region contains a singleindividual element of the data matrix.
 17. The method according to claim16, wherein placing, by the one or more processors, at each identifiedcontrast transition, an intermediate hypersurface comprises placing theintermediate hypersurfaces on the grid structure.
 18. The methodaccording to claim 1, wherein the defined sequence has the form“A”→“C”→“A”→“C”→ . . . , wherein “A” represents an auxiliary point and“C” represents a connecting point.
 19. A computer device configured tocarry out a method for producing at least one boundary hypersurface froma data matrix, the computer device comprising: one or more processorreadable memories configured to store the data matrix and to store datarepresentative of the at least one boundary hypersurface; one or moreprocessors configured to: identify contrast transitions situated betweenrespective pairs of matrix elements; place, at each identified contrasttransition, an intermediate hypersurface; represent each identifiedintermediate hypersurface with a structure including an auxiliary pointand one or more connecting points, wherein the auxiliary point ispositioned on a line that connects the corresponding two respectivematrix elements between which a contrast transition is situated, andwherein the one or more connecting points are each positioned on aboundary with another intermediate hypersurface; generate one or morevector paths that connect, in a defined sequence to form directionallyoriented closed curves that form hypersurface components, auxiliarypoints of intermediate hypersurfaces by extending through connectingpoints between adjacent auxiliary points; and combine hypersurfacecomponents formed by the one or more directionally oriented closedcurves to form at least one boundary hypersurface, wherein the datamatrix has a dimension n and the hypersurface has a dimension n−1;wherein, if the connecting points comprise one or more contact points, acontact point being a connecting point having neighboring matrixelements internal to the hypersurface which contact one another in atmost n−2 dimensions, the one or more processors generate the vectorpaths by selecting one of a connection mode or a separation mode, suchthat in the connection mode, the one or more processors generate vectorpaths to include the contact points in the region internal to thehypersurface, and in the separation mode, the one or more processorsgenerate the vector paths to exclude the contact points from the regioninternal to the hypersurface.